I suppose the thinking was that an eastern stalemate along Belarus/Ukraine; and western invasions stopped on the beaches; there would have been enough resources for an indefinite holdout towards a peace with a remnant of empire.

First, by late, pretty much the entire Luftwaffe was fighting over Germany trying to mitigate the damage from bombing, so there were no air forces to spare. Second, the Germans did transfer a lot of their strength and many of their best units and supplies, to the west. It wasn’t enough, and it was never going to be enough because no matter how many divisions the Germans sent to the west, they were going to get obliterated by allied air power.

But even if we assume that some stalemate could be achieved in the west, what good would that do Germany with Stalin marching in from the east? That still leaves a surrounded Germany being reduced to rubble from the air, increasingly short on food, manpower, oil, and other vital resources.

It is basically the same dynamic we are seeing today in the Ukraine. Absent some real military snafus, the side with the greater active manpower, accessible resources, and domestic production capacity is likely to prevail.

The reason for Sicily was the same as Salerno. Air power cover.

I would probably have taken Sardinia and Corsica and then made a strong landing in Southern France (bigger than operation Dragoon) or in western Italy, above Rome.